Multiple planar coincidences with N-fold symmetry

The coincidence problem for planar patterns with $N$-fold symmetry is considered. For the N-fold symmetric module with $N<46$, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case $N \ge 46$ is briefly discussed and N=46 is described explicitly. The results of the coincidence problem also solve the problem ...

The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N-fold symmetry that emerge as th...

In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in cr...

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to various zeta functions of orders in ima...

Several combinatorial problems of (quasi-)crystallography are reviewed with special emphasis on a unified approach, valid for both crystals and quasicrystals. In particular, we consider planar sublattices, similarity sublattices, coincidence sublattices, their module counterparts, and central and averaged shelling. The corresponding counting functions are encapsulated in Dirichlet series generating functions, with explicit results for the triangular lattice and the twelvefold...

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathema...

The coincidence site lattice (CSL) problem and its generalization to Z-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.

A lattice in Euclidean $d$-space is called well-rounded if it contains $d$ linearly independent vectors of minimal length. This class of lattices is important for various questions, including sphere packing or homology computations. The task of enumerating well-rounded sublattices of a given lattice is of interest already in dimension 2, and has recently been treated by several authors. In this paper, we analyse the question more closely in the spirit of earlier work on simil...

We survey some recent developments in the analytic theory of multiple Dirichlet series with arithmetical coefficients on the numerators.

Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of a lattice $\Gamma$ with a rotated copy $R\Gamma$ of itself. They are useful for classifying grain boundaries and have been studied extensively since the mid sixties. Recently the interests turned to so-called multiple CSLs, i.e. intersections of $n$ rotated copies of a given lattice $\Gamma$, in particular in connection with lattice quantizers. Here we consider multiple CSLs for the 3-dimensional bod...